Published for SISSA by Springer Received: April 18, 2017 Revised: June 7, 2017 Accepted: July 10, 2017 Published: July 17, 2017 Influence of inhomogeneities on holographic mutual JHEP07(2017)082 information and butterfly effect Rong-Gen Cai,a;b Xiao-Xiong Zenga;c and Hai-Qing Zhangd aCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China bSchool of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China cSchool of Material Science and Engineering, Chongqing Jiaotong University, Chongqing 400074, China dDepartment of Space Science and International Research Institute of Multidisciplinary Science, Beihang University, Beijing 100191, China E-mail: [email protected], [email protected], [email protected] Abstract: We study the effect of inhomogeneity, which is induced by the graviton mass in massive gravity, on the mutual information and the chaotic behavior of a 2+1-dimensional field theory from the gauge/gravity duality. When the system is near-homogeneous, the mutual information increases as the graviton mass grows. However, when the system is far from homogeneity, the mutual information decreases as the graviton mass increases. By adding the perturbations of energy into the system, we investigate the dynamical mutual information in the shock wave geometry. We find that the greater perturbations disrupt the mutual information more rapidly, which resembles the butterfly effect in chaos theory. Besides, the greater inhomogeneity reduces the dynamical mutual information more quickly just as in the static case. Keywords: AdS-CFT Correspondence, Black Holes, Classical Theories of Gravity, Ther- mal Field Theory ArXiv ePrint: 1704.03989 Open Access, c The Authors. https://doi.org/10.1007/JHEP07(2017)082 Article funded by SCOAP3. Contents 1 Introduction1 2 Reviews of shock wave geometry and holographic mutual information3 2.1 Shock wave geometry4 2.2 Holographic mutual information5 JHEP07(2017)082 3 Holographic mutual information in the static background case7 3.1 Effect of the width of the strip on the mutual information7 3.2 Effects of the graviton mass and black hole charge on the mutual information9 3.3 The critical width 10 4 Holographic mutual information in the dynamical background case 11 4.1 Black hole charge and butterfly effect 13 4.2 Graviton mass and butterfly effect 15 5 Conclusions and discussions 16 1 Introduction Over the past two decades, there have been a large overlap studies among high energy theory, condensed matter physics, quantum information and geometry based on the con- jecture of gauge/gravity correspondence [1{3]. The typical example is the holographic entanglement entropy (HEE) proposed by Ryu and Takayanagi that relates the minimized codimension-2 bulk surface to the quantum information in the boundary field theory [4]. The Ryu-Takayanagi conjecture greatly simplifies the computations of the entanglement entropy in the field theory, and numerous papers investigates the relation between the bulk geometry and the quantum information in the boundary field theory. For recent re- views, please refer to refs. [5,6 ]. Among these studies, one interesting thing is to adopt the HEE to probe the thermalization process or phase transitions in the strongly coupled systems [7{17]. Mutual information measures the correlations between two subsystems in quantum information theory [18]. The holographic setup of mutual information can be obtained by calculating the length of the wormhole which connects the two sides of an eternal AdS black hole [19{21]. In [20], the authors defined the thermofield double states (TFD) to describe the states of the entangled two sides of the black hole, i.e., β X − Ei jΨi ≡ e 2 jiiL ⊗ jiiR ; (1.1) i { 1 { where β is the inverse of the temperature while jiiL and jiiR are the identical quantum states on the two-sided AdS black holes. Supposing there are two identical space-like subregions A and B on each side of the black hole, the mutual information I(A; B) between A and B can be computed as I(A; B) ≡ S(A) + S(B) − S(A [ B) ; (1.2) where S(A), S(B) are the entanglement entropy of A and B respectively, while S(A[B) is the entanglement entropy of the union of A and B. Holographically, S(A) and S(B) can be calculated by the areas of the minimal surfaces in the bulk geometry associated to A and B independently from the Ryu-Takayanagi conjecture, while S(A [ B) can be computed JHEP07(2017)082 from the minimal surface which crosses the horizon and stretches through the wormhole connecting the two subregions A and B [22]. The disruption of mutual information was related to the butterfly effect by Shenker and Stanford [23]. Specifically, as a small perturbation, such as a light-like energy perturbation, is added to one side of the eternal black hole, the mutual information between the two sides may be disrupted after an amount of time t∗, which means there is no dependence or entanglement between the two sides of the black hole. The time t∗, usually called the scrambling time in black hole systems, is proportional to the logarithm of the entropy of the black hole, e.g., t∗ ∼ β log(S)=2π, in which S is the Bekenstein-Hawking entropy. The disruption of the mutual information is a piece of evidence of the system's high sensitivity to the initial conditions, which reminds us of the terminology in chaos theory-butterfly effect. Recent literatures relevant to the butterfly effect can be found in [24{29]. In this paper, we intend to study the static and dynamical mutual information in the background of massive gravity [30, 31]. As we know from [32{34] that the graviton mass in the bulk breaks the diffeomorphism invariance, which makes the stress-energy tensor non-conserved in the dual field theory. The non-conservation of the stress-energy tensor causes the momentum dissipation in the boundary field theory. Therefore, from this aspect the graviton mass plays the role of inhomogeneity in the boundary field theory, i.e., greater graviton mass is dual to greater inhomogeneity in the boundary.1 There have been a number of papers investigating the inhomogeneous effects caused by massive gravity so far, see for example [35{43]. Firstly, we study the holographic mutual information of two identical strips in a static background of a 3+1-dimensional massive gravity. For the strips with larger length, the mutual information between them decreases monotonically as the graviton mass increases; however, the mutual information between two shorter strips first increases and then de- creases with respect to the graviton mass. A plausible reason is that the graviton mass (or equivalently spatial inhomogeneity) would have greater effects on the strips with larger length. When the system is near-homogeneous (or equivalently with small graviton mass), the mutual information for the strips with shorter length will increase with respect to the 1Strictly speaking, the term \inhomogeneity" used here represents the meaning of translational symmetry breaking or momentum dissipation, which was used previously in the framework of holography in the paper [34]. To be consistent with existing literatures, here we still take the term \inhomogeneity" to represent the same meaning. { 2 { temperature of the black hole (referring to the right panel of figure4 and discussions in the main context), which is similar to the relation between mutual information and tempera- ture in the pure homogeneous case [24]. However, as the graviton mass grows big enough, i.e., far from homogeneity, the mutual information will instead decrease with respect to the graviton mass, which behaves distinctly from the homogenous case. Therefore, we argue that the greater graviton mass (or stronger inhomogeneous effects) would play a dominant role in decreasing the mutual information compared with the temperature. On the other hand, if the strips are longer, the mutual information decreases monotonically according to the graviton mass regardless of the system being near or far from homogeneity. It intuition- aly suggests that the spatial inhomogeneity would have greater impacts to the strips with JHEP07(2017)082 larger length by destroying the mutual information monotonically. A detailed discussion of the relation between mutual information and the graviton mass in the static background case is given in section3 . In all the parameter regimes we considered in this paper, we find that the critical width of the strip, which renders the mutual information vanishing, always decreases according to the graviton mass. Moreover, as the width of the boundary strip increases, it is found that the critical charge which disrupts the mutual information increases as well. As we know, in the massive gravity the temperature of the black hole will decrease as the charge increases. Therefore, we can infer that the critical width of the strip increases when the temperature of the black decreases, which is consistent with the results obtained in [24]. In order to study the dynamical behavior of mutual information in the boundary field theory, one of the approaches is to add extra energy perturbations into the bulk, which will lead to a shift on the horizon of the black hole. This shift will affect the mutual information on the two sides of the black hole. The dynamics of the bulk after adding the perturbations can be conveniently investigated in the shock wave geometry with Kruskal coordinates [44]. We find that as the shift grows (more added perturbed energy at the initial time), the mutual information will be reduced more significantly, which reminds us of the phenomenon in chaos theory | butterfly effect. By turning on the graviton mass, we also find that the greater the mass is, the smaller the mutual information will be, which indicates that the spatial inhomogeneity will reduce the mutual information just like in the static case mentioned above.